About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 289168, 10 pages
http://dx.doi.org/10.1155/2012/289168
Research Article

Uniqueness of Traveling Waves for a Two-Dimensional Bistable Periodic Lattice Dynamical System

Department of Applied Mathematics, National Chung Hsing University, 250 Kuo Kuang Road, Taichung 402, Taiwan

Received 28 December 2011; Accepted 30 January 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Chin-Chin Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. A. Fisher, “The wave of advance of advantageous genes,” Annals of Eugenics, vol. 7, pp. 355–369, 1937.
  2. A. N. Kolmogorov, I. G. Petrovsky, and N. S. Piscounov, “Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,” Bulletin de l'Universite d'Etat à Moscou Serie International section A, vol. 1, pp. 1–25, 1937.
  3. B. Zinner, G. Harris, and W. Hudson, “Traveling wavefronts for the discrete Fisher's equation,” Journal of Differential Equations, vol. 105, no. 1, pp. 46–62, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. S.-C. Fu, J.-S. Guo, and S.-Y. Shieh, “Traveling wave solutions for some discrete quasilinear parabolic equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 48, no. 8, pp. 1137–1149, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. X. Chen and J.-S. Guo, “Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations,” Journal of Differential Equations, vol. 184, no. 2, pp. 549–569, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. X. Chen and J.-S. Guo, “Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics,” Mathematische Annalen, vol. 326, no. 1, pp. 123–146, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. J. Carr and A. Chmaj, “Uniqueness of travelling waves for nonlocal monostable equations,” Proceedings of the American Mathematical Society, vol. 132, no. 8, pp. 2433–2439, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. X. Chen, S.-C. Fu, and J.-S. Guo, “Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices,” SIAM Journal on Mathematical Analysis, vol. 38, no. 1, pp. 233–258, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J.-S. Guo and F. Hamel, “Front propagation for discrete periodic monostable equations,” Mathematische Annalen, vol. 335, no. 3, pp. 489–525, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. B. Zinner, “Stability of traveling wavefronts for the discrete Nagumo equation,” SIAM Journal on Mathematical Analysis, vol. 22, no. 4, pp. 1016–1020, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. B. Zinner, “Existence of traveling wavefront solutions for the discrete Nagumo equation,” Journal of Differential Equations, vol. 96, no. 1, pp. 1–27, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. S.-N. Chow, J. Mallet-Paret, and W. Shen, “Traveling waves in lattice dynamical systems,” Journal of Differential Equations, vol. 149, no. 2, pp. 248–291, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. J. Mallet-Paret, “The Fredholm alternative for functional-differential equations of mixed type,” Journal of Dynamics and Differential Equations, vol. 11, no. 1, pp. 1–47, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J. Mallet-Paret, “The global structure of traveling waves in spatially discrete dynamical systems,” Journal of Dynamics and Differential Equations, vol. 11, no. 1, pp. 49–127, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. P. W. Bates, X. Chen, and A. J. J. Chmaj, “Traveling waves of bistable dynamics on a lattice,” SIAM Journal on Mathematical Analysis, vol. 35, no. 2, pp. 520–546, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. X. Chen, J.-S. Guo, and C.-C. Wu, “Traveling waves in discrete periodic media for bistable dynamics,” Archive for Rational Mechanics and Analysis, vol. 189, no. 2, pp. 189–236, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. J. W. Cahn, J. Mallet-Paret, and E. S. Van Vleck, “Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice,” SIAM Journal on Applied Mathematics, vol. 59, no. 2, pp. 455–493, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. J.-S. Guo and C.-H. Wu, “Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system,” Osaka Journal of Mathematics, vol. 45, no. 2, pp. 327–346, 2008. View at Zentralblatt MATH
  19. J.-S. Guo and C.-H. Wu, “Front propagation for a two-dimensional periodic monostable lattice dynamical system,” Discrete and Continuous Dynamical Systems, vol. 26, no. 1, pp. 197–223, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. P. C. Fife and J. B. McLeod, “The approach of solutions of nonlinear diffusion equations to travelling front solutions,” Archive for Rational Mechanics and Analysis, vol. 65, no. 4, pp. 335–361, 1977. View at Zentralblatt MATH
  21. J. P. Keener, “Propagation and its failure in coupled systems of discrete excitable cells,” SIAM Journal on Applied Mathematics, vol. 47, no. 3, pp. 556–572, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. S.-G. Liao and C.-C. Wu, “Propagation failure in discrete periodic media,” Preprint. In press.